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The stability of stratified spatially periodic shear flows at low Péclet number

Garaud, Pascale and Gallet, Basile and Bischoff, Tobias (2015) The stability of stratified spatially periodic shear flows at low Péclet number. Physics of Fluids, 27 (8). Art. No. 084104. ISSN 1070-6631.

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This work addresses the question of the stability of stratified, spatially periodic shear flows at low Péclet number but high Reynolds number. This little-studied limit is motivated by astrophysical systems, where the Prandtl number is often very small. Furthermore, it can be studied using a reduced set of “low-Péclet-number equations” proposed by Lignières [“The small-Péclet-number approximation in stellar radiative zones,” Astron. Astrophys. 348, 933–939 (1999)]. Through a linear stability analysis, we first determine the conditions for instability to infinitesimal perturbations. We formally extend Squire’s theorem to the low-Péclet-number equations, which shows that the first unstable mode is always two-dimensional. We then perform an energy stability analysis of the low-Péclet-number equations and prove that for a given value of the Reynolds number, above a critical strength of the stratification, any smooth periodic shear flow is stable to perturbations of arbitrary amplitude. In that parameter regime, the flow can only be laminar and turbulent mixing does not take place. Finding that the conditions for linear and energy stability are different, we thus identify a region in parameter space where finite-amplitude instabilities could exist. Using direct numerical simulations, we indeed find that the system is subject to such finite-amplitude instabilities. We determine numerically how far into the linearly stable region of parameter space turbulence can be sustained.

Item Type:Article
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URLURL TypeDescription Paper
Bischoff, Tobias0000-0003-3930-2762
Additional Information:© 2015 AIP Publishing LLC. Received 1 February 2015; accepted 22 July 2015; published online 14 August 2015. This work was initiated as a project at the Woods Hole GFD summer program in 2013. The authors thank the program for giving them the opportunity to collaborate on this topic and for their financial support. P.G. was also funded by NSF No. CAREER-0847477 and by NSF No. AST 1517927. B.G. was funded by the junior grant TURBA from Labex PALM No. ANR-10-LABX-0039. All simulations presented here were performed on the Hyades computer, purchased at UCSC with an NSF MRI grant. The authors thank S. Stellmach for providing his code. The authors also thank C. Caulfield, C. Doering, R. Kerswell, and S. Stellmach for their help and for inspiring discussions.
Funding AgencyGrant Number
Agence Nationale pour la Recherche (ANR)ANR-10-LABX-0039
Issue or Number:8
Record Number:CaltechAUTHORS:20150924-160530478
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Official Citation:The stability of stratified spatially periodic shear flows at low Péclet number Garaud, Pascale and Gallet, Basile and Bischoff, Tobias, Physics of Fluids, 27, 084104 (2015), DOI:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:60499
Deposited By: Tony Diaz
Deposited On:24 Sep 2015 23:28
Last Modified:03 Oct 2019 08:57

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